Theoretical key

Impossible and theoretical keys are really one and the same; basically, they are keys that have no possible conventional key signature.

For example the key of D flat minor cannot logically exist, as its key signature would have to contain a double flat - an impossibility in conventional notation. In equal temperament, the (real) key of C sharp minor is the only possible way that this key can be notated, as C sharp logically equals D flat. In the same way, there are 5 keys that are duplicated by this relationship:

1. C sharp and D flat majors
2. D sharp and E flat minors
3. F sharp and G flat majors
4. G sharp and A flat minors
5. A sharp and B flat minors

In the same way that e.g. C sharp equals D flat, C sharp also sounds exactly the same as D flat; hence C sharp major sounds exactly the same as D flat major on the ear.

The difference between these two is only distinguished by the obvious difference in appearance on a page of printed music, but also the implications of such a key as a secondary key to the primary key of the piece of music in question; that is to say that if a piece in F minor were to modulate to its sub-mediant, D flat major, no confusion would arise as whether to use C sharp or D flat majors: F minor and D flat major both have flats in their key signatures anyway.

These duplicated keys, having 5, 6 or 7 sharps/flats in their key signatures, are the most "distant" keys to use a conventional key signature. In keys beyond these, double-sharps and double-flats have to be incorperated into the key signature; the following 6 keys require 1, 2 or 3 double-sharps or double-flats; these 6 keys are the parallel major/minor keys of the 6 duplicated keys above:

1. D flat minor; 8 flats (=C sharp minor) relative key: F flat major
2. D sharp major; 9 sharps (=E flat major) relative key: B sharp minor
3. G flat minor; 9 flats (=F sharp minor) relative key: B double-flat major
4. G sharp major; 8 sharps (=A flat major) relative key: E sharp minor
5. A sharp major; 10 sharps (=B flat major) relative key: F double-sharp minor
6. C flat minor; 10 flats (=B minor) relative key: E double-flat major

However, these keys, along with all keys for that matter, can exist in a different form using the 12 modes as a basis for their notation in a (modal) key signature e.g. the key of D flat minor could exist in the Dorian mode, using the B flat to start with in the key signature and taking the form of the key signature of A flat minor.

With regular 12-tone tuning, these keys, and all other "impossible" ones, can be re-spelled as conventional keys. However, in a different tuning system (for example, 19 tone equal temperament), several keys do require a double-sharp or double-flat in the key signature, and no longer have conventional equivalents: where there are 19 tones in the scale, the key of B double-flat major (9 flats) is equivalent to A sharp major (10 sharps).

Adding or removing 12 sharps or flats (or the same number as the tuning system) to or from a key signature results in the same key enharmonically respelled. Here are the six simplest keys respelled as "impossible" keys (with 11, 12 or 13 sharps or flats), each in two different ways:

1. C major = B sharp major or D double-flat major
2. D minor = C double-sharp minor or E double-flat minor
3. E minor = D double-sharp minor or F flat minor
4. F major = E sharp major or G double-flat major
5. G major = F double-sharp major or A double-flat major
6. A minor = G double-sharp minor or B double-flat minor